Best Known (32−8, 32, s)-Nets in Base 9
(32−8, 32, 3283)-Net over F9 — Constructive and digital
Digital (24, 32, 3283)-net over F9, using
- net defined by OOA [i] based on linear OOA(932, 3283, F9, 8, 8) (dual of [(3283, 8), 26232, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(932, 13132, F9, 8) (dual of [13132, 13100, 9]-code), using
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(932, 13132, F9, 8) (dual of [13132, 13100, 9]-code), using
(32−8, 32, 13132)-Net over F9 — Digital
Digital (24, 32, 13132)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(932, 13132, F9, 8) (dual of [13132, 13100, 9]-code), using
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
(32−8, 32, large)-Net in Base 9 — Upper bound on s
There is no (24, 32, large)-net in base 9, because
- 6 times m-reduction [i] would yield (24, 26, large)-net in base 9, but